Question

The order and degree of the differential equation $\frac{d^{2}y}{dx^{2}}+\sqrt{x^{2}+\left(\frac{dy}{dx}\right)^{3/2}}=0$ are respectively

• A. 2, 4
• B. 2, 3
• C. 2, 2
• D. 3, 4
• E. 4, 3

Hint:

Calculus Tip: You must clear ALL fractional powers on ANY derivative (even lower-order ones) before you can determine the official degree of the equation!

Answer 1

The Correct Option is A

Concept:
The order of a differential equation is the highest derivative present. The degree is the highest power of that highest derivative, but only after the equation has been cleared of all fractional powers (radicals) concerning the derivatives.

Step 1: Determine the order.
The highest derivative in the equation is $\frac{d^2y}{dx^2}$ (the second derivative). Therefore, the order is exactly $2$.

Step 2: Isolate the radical term.
To find the degree, we must remove the square root and the fractional power. First, move the radical to the other side: $$\frac{d^2y}{dx^2} = -\sqrt{x^2 + \left(\frac{dy}{dx}\right)^{3/2}}$$

Step 3: Square both sides to remove the root.
Square both sides of the equation: $$\left(\frac{d^2y}{dx^2}\right)^2 = x^2 + \left(\frac{dy}{dx}\right)^{3/2}$$

Step 4: Isolate the remaining fractional power.
Move the $x^2$ term over to isolate the $3/2$ power on the first derivative: $$\left(\frac{d^2y}{dx^2}\right)^2 - x^2 = \left(\frac{dy}{dx}\right)^{3/2}$$

Step 5: Square both sides again to clear all fractions.
Square the entire equation to remove the $/2$ denominator in the exponent: $$\left[ \left(\frac{d^2y}{dx^2}\right)^2 - x^2 \right]^2 = \left(\frac{dy}{dx}\right)^3$$ If we expand the left side, the leading term is $\left(\frac{d^2y}{dx^2}\right)^4$. The highest power of the highest-order derivative is now 4. So, the degree is 4. Hence the correct answer is (A) 2, 4.